PSCI 2702 Quantitative Research Methods Lecture 5

Quantitative Research Methods in Political Science

• This course covers lecture 5, focusing on inferential statistics, specifically sampling and sampling distributions.
• The instructor is Michael E. Campbell, and the course number is PSCI 2702 (A).
• The date of the lecture is 10/03/2024.

A Note on Symbols

• Statistical symbols used to represent statistics differ depending on whether working with a sample or a population.
• A table (Table 5.4) details symbols for means and standard deviations for samples, populations and sampling distributions of means and proportions.

Descriptive vs. Inferential Statistics

• Descriptive statistics summarize data or describe the distribution of a single variable (univariate) or describe relationships between multiple variables (bivariate or multivariate).
• Inferential statistics allow generalization from samples to populations. They use data from a carefully chosen sample subset to make inferences about a larger population.

Samples and Sampling

• Sampling involves taking a group of cases drawn from a larger population.
• Examples include determining characteristics of the entire Canadian population, or characteristics about Canadian university students.
• Surveying every individual in large populations is typically too expensive or time-consuming.

Samples and Sampling Cont'd

• Populations are often too large to study every individual.
• The solution to this issue is to sample a carefully selected subset of the population.
• Samples must accurately represent the population for accurate results to be achieved.

Sampling and Samples Cont'd

• Two main types of sampling are presented:
  • Non-probability sampling: used when representing the entire population is unnecessary or resources are limited.
  • Probability sampling: used when representing the entire population is necessary (for example, determining if respect for politicians affects voter turnout in democratic countries).

Non-Probability Sampling

• Non-probability sampling techniques are used when researchers are not concerned about representing the whole population, or when resources are limited.
• Types include:
  • Convenience sampling: choosing individuals easily accessible to the researcher (e.g., a study of Montreal residents by a researcher in Montreal).
  • Snowball sampling: used when researchers want to reach difficult-to-access populations (e.g., those in conflict zones or drug cartels).
  • Quota sampling: selecting a sample representing the population in specified sub-categories, but not random (e.g., selecting a specific income bracket).

Representativeness

• A sample is considered representative if it reflects the important characteristics of the population.
• To achieve a representative sample, probability sampling techniques are used.

Probability Sampling

• Probability sampling, also known as random sampling, uses careful techniques to ensure every case has a chance of being chosen.
• A poor example would be selecting the first 1000 people leaving a grocery store for a survey.
• A good example of using random sampling is using a table of random numbers.

Probability Sampling Cont'd

• The goal of probability sampling is to achieve representativeness in sample selection, with the sample mirroring the population's characteristics.
• Traits and proportions of the population should be reflected in the sample regardless of size.

EPSEM

• EPSEM (Equal Probability of Selection Method) ensures every person in a population has an equal chance of being selected for a sample.
• This is the foundation of simple random sampling.
• Refinements to sampling procedures (e.g., systematic random samples, stratified random samples, and cluster sampling)are sometimes used.

Selection Process for Simple Random Sample

• To obtain a simple random sample, all members of the population are listed and then assigned a unique ID number.
• Random selection can be done using a table of random numbers.
• A table of random numbers can be employed to obtain a simple random sample (i.e., EPSEM).

Selection Process for a Random Sample Using Table of Random Numbers

• Cases for a sample can be selected using a table of random numbers that corresponds to a population's unique case numbers.
• Assign unique IDs to each case in the population list.
• Select participants from the table; each random number matching a case ID would be chosen.

Sampling Distribution

• When working with samples, there is typically no information available about the population.
• Inferential statistics use information from a sample to learn about the population.

Sampling Distribution Cont'd

• To characterize a variable, three types of information are necessary: The shape of the distribution, some measure of central tendency, and some measure of dispersion.
• All these could be obtained from a sample.

Sampling Distribution Cont'd

• The sampling distribution shows the theoretical, probabilistic distribution of a statistic for all possible samples of a certain sample size.

Sampling Distribution Cont'd

• There are three distinct distributions in inferential statistics: the sample distribution, the population distribution, and the sampling distribution.

Sampling Distribution Cont'd

• Sampling distributions allow for the estimation of probabilities for any particular sample outcome.
• Just like the normal curve, sampling distributions are theoretical constructs.

Constructing Sampling Distribution Example #1

• Researchers want to understand the age in a small town.
• A sample of 100 people is selected using EPSEM to find an average age of 27.
• This is a single sample, and many more could be obtained.

Constructing Sampling Distribution Example #1 Cont'd

• An infinite number of samples can be obtained to estimate the mean for a population.
• In a second sample, a mean age of 30 may be found, showing how sample means vary.

Constructing Sampling Distribution Example #1 Cont'd

• In a distribution of sample means, with an infinite number of samples, the central limit theorem states that the mean is likely to be found close to the population mean.
• This means the majority of sample means will likely be clustered near the population mean.

Constructing Sampling Distribution Example #1 Cont'd

• The number of sample means that will be close to the mean in the population—and those that might fall far away—will follow a normal curve.

Constructing Sampling Distribution Example #2

• A population of four people (N=4) is used to illustrate the sampling distribution of sample means (n=2).
• This population has varying income values ($2, $4, $6, $8).
• The average (mean) of the population is $5.
• The standard deviation (σ) is calculated as $2.24.

Constructing Sampling Distribution Example #2 Cont'd

• Samples of two people each (n=2) are taken from the population by drawing with replacement.
• The average (mean) of sample means (mean of means) will likely be $5.
• The standard deviation of sample means (standard error) will be $1.58 (the standard deviation divided by the square root of the sample size).

Constructing Sampling Distribution Example #2 Cont'd

• The means for samples of size 2 (n=2) have a normal distribution.
• This normal distribution is demonstrated visually in a histogram.

Constructing Sampling Distribution Example #2 Cont'd

• The mean of the sampling distribution has the same mean as the population (shown in Table 5.3)
• The standard deviation of the sampling distribution (or standard error) is equal to the population standard deviation divided by the square root of the sample size.

Standard Error

• The smaller the standard error, the more representative the sample is.
• Larger samples provide a more accurate representation of the population.
• The standard error reflects the expected deviation of the sample mean from the population mean.

Linking the Population, the Sampling Distribution, and the Population Review

• Inferential statistics link samples to populations, and these links are based on sample characteristics' data collection.
• A random sample using EPSEM is necessary for data collection.
• Sample traits are gathered, and population data is not required.

Linking the Population, the Sampling Distribution, and the Population Review Cont'd

• Theorems provide knowledge of the sampling distribution.
• Sample sizes larger than 100 produce distributions that are similar to the population's distribution.

This quiz covers Lecture 5 of PSCI 2702, focusing on inferential statistics, particularly sampling and sampling distributions. You'll explore the differences between descriptive and inferential statistics, along with the relevant statistical symbols. Test your understanding of these key concepts in quantitative research methods.